MODIFICATION OF SIMPLEX METHOD AND ITS IMPLEMENTATION IN VISUAL BASIC. Nebojša V. Stojković, Predrag S. Stanimirović and Marko D.

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1 MODIFICATION OF SIMPLEX METHOD AND ITS IMPLEMENTATION IN VISUAL BASIC Nebojša V Stojković, Predrag S Stanimirović and Marko D Petković Abstract We investigate the problem of finding the first basic solution in the simplex algorithm A modification of two phases simplex method is presented and its implementation in programming language Visual Basic is described A few Netlib examples are presented Key words: Linear programming, simplex method, Visual Basic Consider the linear program 1 Introduction (11) Maximize f(x) = f(x 1,, x n ) = subject to N (1) i N (2) i J i c i x i d i=1 a ij x N,j b i (i = 1,, p) a ij x N,j b i (i = p + 1,, q) a ij x N,j = b i (i = q + 1,, m) x i 0, (i = 1,, m) 1991 Mathematics Subject Classification 90C05 1 Typeset by AMS-TEX

2 2 Nebojša V Stojković, Predrag S Stanimirović and Marko D Petković We can transform constraints (11) in the following way Every inequality of the form N (1) i we change with the equality N (1) i a ij x N,j b i = x B,i (i = 1,, p), and every inequality of the form N (2) i we change with the equality N (2) i a ij x N,j b i = x B,i (i = p + 1,, q) In a such way we get the linear program Max c 1 x c n x n d (12) Ax = b, b = (b 1,, b m ), x i 0, (i = 1,, n + q), where the matrix A is in R m (n+q) We suppose that the system is of full rank ( rank(a) = m) After the substitution n + q m = n, the canonical form of problem (12) is written in the following tableau form [1], [6] (13) x N,1 x N,2 x N,n 1 a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn = b 1 = b 2 = b m = x B,1 = x B,2 = x B,m c 1 c 2 c n d = f where x N,1,, x N,n is the set of nonbasic variables and x B,1,, x B,m are basic variables Transformed coefficients of the matrix A and the vector c are denoted by a ij and c j, respectively, for the sake of simplicity The paper is organized as follows In the second section we introduce the modification of two phases simplex method and investigate the problem of finding the first basic solution in the simplex algorithm In order to accelerate the process for generating the first basic feasible solution we investigate two problems: - the problem of the first choice of basic and nonbasic variables, and - the problem of the replacement of a basic and a nonbasic variable in the general simplex method In the third section we describe an implementation of the two phases simplex method and its modification in the programming language Visual Basic Several numerical examples are reported in the last section

3 Modification of simplex method and its implementation in visual basic 3 2 Modification For the sake of completeness we restate one version of the classical two phases maximization algorithms from [3] and [7] with respect to linear problem (11), which is presented in the tableau form (13) Algorithm 1 Replacing a basis variable x B,p and nonbasis variable x N,j a 1 pj = 1, a 1 pl = a pl, l j, b 1 p = b p, a 1 ql = a qi, q p, a 1 ql = a ql a pla qj, q p, l j, c 1 l = c l c ja pl, l j, c 1 j = c j, d 1 = d b pc j Algorithm 2 (Simplex method for basic feasible solution) Step S1A If c 1,, c n 0, then the basic solution is an optimal solution Step S1B Choose an arbitrary c j > 0 (We use the maximal c j ) Step S1C If a 1j,, a mj 0, stop the algorithm Maximum is + Otherwise, go to the next step Step S1D Compute { } bi min a ij > 0 = b p 1 i m a ij and replace nonbasic and basic variables x N,j and x B,p, respectively, applying Algorithm 1 If the condition b 1,, b m 0 is not satisfied, then we use the following steps to search for the first basic feasible solution Algorithm 3 Step S2 Select the last b i < 0

4 4 Nebojša V Stojković, Predrag S Stanimirović and Marko D Petković Step S3 If a i1,, a in 0 then STOP Linear program can not be solved Step S4 Otherwise, find a ij < 0, compute ({ } { }) bi bk min a kj > 0 = b p k>i a ij a kj and replace nonbasic and basic variables x N,j and x B,p, respectively We use the last a ij < 0 In the begining we suppose that the matrix A is of full rank ( rank(a) = m) This is valid if the equalities in J i are lineary independent Otherwise, we apply Gauss-Jordan algorithm for the elimination of dependent equalities After that, we apply the next algorithm to obtain the canonical form (13) Algorithm 4 Step 1 If J i =, perform Algorithm 3 Step 2 Find the first i such that ith constraint is a equality and choose the last a ij 0 for the pivot element (If there not exists a ij 0 and b i 0, the problem is not feasible; if b i = 0, then we can drop ith constraint) Step 3 Apply Algorithm 1 and drop the jth column Step 4 Find the next i such that the ith constraint is an equality and perform Step 2 Otherwise, apply Algorithm 3 The problem of the replacement of a basic and a nonbasic variable in the general simplex method is contained in Step S1D and Step S4 We observed two drawbacks of Step S4 1 If p = i and if there exists index t < i = p such that b t a tj < b p, b t > 0, a tj > 0 in the next iteration x B,t becomes negative: x B,t = b 1 t = b t b p a tj < 0 2 If p>i, in the next iteration b 1 i is negative, but there may exists b t <0, t < i such that ({ } { }) bt bk min, a tj > 0 a kj < 0, b k > 0 = b t k>t a tj a kj a tj

5 Modification of simplex method and its implementation in visual basic 5 In this case, it is possible to choose a tj for the pivot element and obtain Also, since x B,t = b 1 t = b t a tj 0 b t a tj b k a kj, each b k > 0 remains convenient for the basic feasible solution: x B,k = b 1 k = b k b t a tj a kj 0 For this purpose, we propose a modification of Step S4 This modification follows from the following lemma Lemma 21 Let the problem (11) be feasible and let x be the basic infeasible solution with b i1,, b iq < 0 Denote I = {i 1,, i q } In the following two cases a) q = m, and b) q < m and there exists r I and s {1,, n} such that { } bh (21) min a hs > 0 b r, < 0, h/ I a hs it is possible to produce the new basic solution x 1 = {x 1 B,1,, x1 B,m } with at most q 1 negative coordinates in only one iterative step of the simplex method, if we choose for the pivot element, ie replace nonbasic variable x N,s with the basic variable x B,r Proof a) If q = m, choose an arbitrary coefficient a ms < 0 for the pivot element Applying the simplex method we get a new solution with at least one coordinate positive For example, we have x 1 B,m = b 1 m = b m a ms 0 b) Assume now that the conditions q < m and (21) are satisfied Choose for the pivot element In this case the coordinates of the new basic solution are x 1 B,i = b 1 i = b i b r a is, i r, x 1 B,r = b 1 r = b r

6 6 Nebojša V Stojković, Predrag S Stanimirović and Marko D Petković For k r, k / I and a ks < 0 it is obvious that For a ks > 0 and k / I, using x 1 B,k = b k b r a ks 0 we conclude immediately b k a ks b r x 1 B,k = b 1 k = b k b r a ks 0 which completes the proof Remark 21 If there is no r such that the condition (21) is valid, let r / I and s {1,, n} be such that { } bh min a hs < 0 = b r h/ I a hs For such r and s we use for the pivot element Applying the simplex transformation we get the new solution x 1 with nonnegative coordinates x 1 B,i = b 1 i = b i b r a is, i / I As the set of the basic solutions is finite, after finitely many transformations, applying anticycling rules if it is necessary, condition (21) will be valid In accordance with these considerations, we propose the following improvement of Algorithm 3 Algorithm 5 Step 1 If b 1,, b m 0 perform Algorithm 2 Otherwise, construct the set B = {b i1,, b iq } = {b ik b ik < 0, k = 1,, q} Step 2 Select the first b is < 0 Step 3 If a is,1,, a is,n 0 then STOP Linear program is not solvable

7 Modification of simplex method and its implementation in visual basic 7 Otherwise, construct the set Q = {a is,j p < 0, p = 1,, t}, set p = 1 and continue Step 4 Compute Step 5 If min 1 k m { } bk a k,jp > 0, b k > 0 = b h a k,jp a p,jp b is a is,j p b h a p,jp then replace nonbasic and basic variables x N,jp and x B,is, else go to Step 6 Step 6 If p > t replace x N,jp and x B,h and go to Step 2 Otherwise, put p = p + 1 and go to Step 3 In the sequel we propose an improvement of Algorithm 1 Observe that in the real problems the matrix A is very sparse, so the number of coeficients needed to transform is not very great Algorithm 6 (The modification of Algortihm 1) Step 1 Form the sets V = {a pl a pl 0, l = 1,, n + 1}, K = {a qj a qj 0, q = 1,, m + 1} Step 2 Apply Algorithm 1 only for a pl, a qj and a ql satisfying a pl V and a qj K The next improvement is in Algorithm 4 Instead dropping columns, we will mark this column In other words, we introduce logical sequence outc with values outc(i) = true if the ith column is not dropped, and outc(i) = f alse otherwise Similary, outr is an indicator for rows Algorithm 7 The improvement of Algorithm 4 Step 1 Set outc(i) = true, i = 1,, n, and outr(j) = true, i = 1,, m Step 2 If J i =, go to Step 7 Step 3 If the ith constraint is the equality, continue Step 4 Find a ij 0 such that outc(j) = true (If there not exists a ij 0 with outc(j) = true and b i 0, the problem is not feasible; if b i = 0, then we can drop the ith constraint and set outr(i) = false)

8 8 Nebojša V Stojković, Predrag S Stanimirović and Marko D Petković Step 5 Put a ij for a pivot element and perform Algortihm 1 Step 6 Go to Step 2 Step 7 Drop all columns and rows with outc(i) = false and outr(j) = false, respectively, set outc(i) = true and outr(j) = true for all i, j and perform Algortihm 3 3 Implementation Simplex algorithm (Algorithms 1, 2, 3 and 4), modification (Algorithm 5) and both improvements (Algorithms 6 and 7) are implemented in the code M arp lex written in the programming language Visual Basic [5] 4 Numerical experience Example 41 We tested the code MarP lex on some Netlib test problems We compare our results with the popular robust code P Cx [2] From Table 1 we can see that our results are in accordance with the results obtained by P Cx Let us mentioned that the code P Cx is not based on the simplex method Instead the simplex algorithm, the interior point primal dual method is implemented in P Cx Problem P Cx MarP lex Algorithm5 Algorithm2 No5 No2 Adlittle Afiro Agg Agg Blend Lotfi Sc Sc Sc50a Sc50b Scagr Scagr Scorpion Sctap Share2b Stocfor LitVera Table 1 Note that our result for the ill-conditioned problem LitV era from [4] is correct and that code P Cx is unable to give better precision in that case In the next two columns we give the number of iterations needed for the first basis feasible solution As we can see, Algorithm 5 is faster then Algorithm 2 almost in all casses In the last two columns the number of all iterations are given References [1] BD Bounday, Basic linear programming, Edvard Arnold, Baltimore, 1984

9 Modification of simplex method and its implementation in visual basic 9 [2] J Czyzyk, S Mehrotra and SJ Wright, PCx User Guide, Optimization Thechnology Center, Technical Report 96/01, 1996 [3] JP Ignizio, Linear programming in single-multiple-objective systems, Englewood Cliffs:Prentice Hall, 1982 [4] V Kovačević-Vujčić, Ill conditioness and interior point method, Technical Report , Laboratory of Operations Research, Faculty of Organizational Sciences, November 1998 [5] Microsoft Developer Network, Microsoft Corporation Inc1998 [6] M Sakarovitch, Linear programming, Springer-Verlag, New York, 1983 [7] S Vukadinović, S Cvejić, Matematičko programiranje, Univerzitet u Prištini, Priština 1996, 1983 (In Serbian)